Proof that log 2 is an irrational number
Before doing the proof, let us recall two things: (1) rational numbers are numbers that can be expressed as where and are integers, and not equal to ; and (2) for any positive real number , its logarithm to base is defined to be a number such that . In proving the statement, we use proof by contradiction.
Theorem: log 2 is irrational
Assuming that log 2 is a rational number. Then it can be expressed as with and are positive integers (Why?). Then, the equation is equivalent to . Raising both sides of the equation to , we have . This implies that . Notice that this equation cannot hold (by the Fundamental Theorem of Arithmetic) because is an integer that is not divisible by 5 for any , while is divisible by 5. This means that log 2 cannot be expressed as and is therefore irrational which is what we want to show.